Page 331 - Advanced Linear Algebra
P. 331
Metric Spaces 315
Example 12.14 The metric space is complete. To prove this, let ² M % ³ be a
Cauchy sequence in , where
M
% ~ ²% Á Á% Á 2 Áó
Then, for each coordinate position ,
B
( c Á Á (% ( % c Á % Á (% ~ ² % Á % ³ ¦
~
which shows that the sequence ²% ³ of th coordinates is a Cauchy sequence in
Á
s
s ( d or ) . Since ( d or ) is complete, we have
²% ³ ¦ & as ¦ B
Á
We want to show that &~ ²& ³ M and that ²% ³ ¦ & .
To this end, observe that for any , there is an for which
5
Á 5 ¬ ( % Á c % Á (
~
for all . Now we let ¦B , to get
5 ¬ ( % Á c & (
~
for all . Letting ¦B , we get, for any 5 ,
B
( % & ( Á c
~
which implies that ²% ³ c & M and so & ~ & c ²% ³ b ²% ³ M and in
addition, ²% ³ ¦ & .
As we will see in the next chapter, the property of completeness plays a major
role in the theory of inner product spaces. Inner product spaces for which the
induced metric space is complete are called Hilbert spaces .
Isometries
A function between two metric spaces that preserves distance is called an
isometry. Here is the formal definition.
Z
Z
Definition Let ²4Á ³ and ²4 Á ³ be metric spaces. A function ¢ 4 ¦ 4 Z is
called an isometry if
Z
² ²%³Á ²&³³ ~ ²%Á &³
